HINTS FOR PROBLEM SET #22

 

Problem Set #22 – Problem 1

Since we are given information about the angular velocity and a time interval, we will use the Principle of Impulse and Momentum the rigid body:

Linear momentum of the body at time 1 + Impulses acting on the body during the time interval (t1 to t2) = Linear momentum of the body at time 2

And:

Angular momentum of the body at time 1 + Angular impulses acting on the body during the time interval (t1 to t2) = angular momentum of the body at time 2

t1 is when the sphere has angular velocity of 90 radians/second.

t2 is when the sphere has come to rest.

To apply the two equations above, draw a diagram consisting of these 3 elements:

     1)  A picture of the sphere with the linear momentum and angular momentum vectors at t1.

2)  A picture of the FBD of the sphere showing the forces as impulses. 

     (Just multiply each force by delta t.)

3)  A picture of the sphere with the linear momentum and angular momentum vectors at t2.

And of course you should put them together in such a way that it is clear that:

the first picture + the second picture = the third picture

The linear momentum vector in pictures 1 and 3 is shown by a vector acting at the mass center and consisting of:  

(mass) x (velocity of the mass center).  

In this case it is zero for both t1 and t2 since the sphere is rotating only.

The angular momentum vector in pictures 1 and 3 is shown by the couple consisting of:

(mass moment of inertia about the mass center of the sphere) x (angular velocity of the sphere).  

In this case it is (2/5 m r^2) x (90) for t1 and 0 for t2.

The forces in the FBD (actually the impulses) will include the weight of the sphere and the frictional and normal forces acting at the wall and floor.  

Note that the frictional forces are kinetic friction.  

Hence: 

       ffloor = (coefficient of kinetic friction) Nfloor

and

       fwall = (coefficient of kinetic friction) Nwall

Now get the x and y components of the linear momentum equation.

Now get the k component of the angular momentum equation.  

These 3 equations will allow you to solve for Nwall, Nfloor and delta t.

 

Problem Set #22 – Problem 2

Since we are interested in information about the velocity and a time interval, we will use the Principle of Impulse and Momentum the rigid body:

Linear momentum of the body at time 1 + Impulses acting on the body during the time interval (t1 to t2) = Linear momentum of the body at time 2

And:

Angular momentum of the body at time 1 + Angular impulses acting on the body during the time interval (t1 to t2) = angular momentum of the body at time 2

t1 is when the double pulley is at rest.

t2 is 1.5 seconds later.

delta t = 1.5 seconds

To apply the two equations above, draw a diagram consisting of these 3 elements:

     1)  A picture of the pulley with the linear momentum and angular momentum vectors at t1.

     2)  A picture of the FBD of the pulley showing the forces as impulses. 

(Just multiply each force by delta t.)

     3)  A picture of the pulley with the linear momentum and angular momentum vectors at t2.

And of course you should put them together in such a way that it is clear that:

the first picture + the second picture = the third picture

The linear momentum vector in pictures 1 and 3 is shown by a vector acting at the mass center and consisting of:  

(mass) x (velocity of the mass center).  

In this case it is zero for t1 and a vector with only an upward component for t2.

The angular momentum vector in pictures 1 and 3 is shown by the couple consisting of:

(mass moment of inertia about the mass center of the pulley) x (angular velocity of the pulley).  

In this case it is 0 for t1 and Iw for t2.

The forces in the FBD (actually the impulses) will include the weight of the sphere, the force P pulling at point B, and the tension T in the cord at C.

Now get y component of the linear momentum equation.  (There will be no x component.)

Now get the k component of the angular momentum equation. 

One more equation is obtained when we note that there is no slipping:

(angular velocity of the pulley) = (velocity of the mass center) / (radius of the pulley), 

These three equations will allow you to solve for v, w and T.